In the euclidean space setting, symmetrization inequalities have been quite useful in solving problems coming from various parts of analysis: spectral geometry, variational problems, mathematical physics, spectral theory, to name a few. This theory is very classical and a lot is known in the continuum. In my talk, I will discuss about discrete analogues of these inequalities on graphs. This is a fairly new topic, and there is hardly anything known in this setting. I will talk about recent developments that has happened in the last one year, connections of this theory with different types of isoperimetric inequalities on graphs. Apart from general theory, I will present some concrete results for standard graph on 2d integer lattices. The talk will be at the interface of discrete math and analysis. It is based on a joint work with Stefan Steinerberger: arXiv:2212.07590.